\(CMR:\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{47}+\sqrt{48}}>3\)
Chứng minh: \(\frac{1}{2\cdot\sqrt{1}}+\frac{1}{3\cdot\sqrt{2}}+\frac{1}{4\cdot\sqrt{3}}+...+\frac{1}{2012\cdot\sqrt{2011}}+\frac{1}{2013\cdot\sqrt{2012}}\)\(< 2\)
Chứng minh: A=\(\frac{1}{3\cdot\left(\sqrt{1}+\sqrt{2}\right)}+\frac{1}{5\cdot\left(\sqrt{2}+\sqrt{3}\right)}+...+\frac{1}{97\cdot\left(\sqrt{48}+\sqrt{49}\right)}\)\(< \frac{1}{2}\)
Bài 1: Tính giá trị của biểu thức:\(\frac{1}{1\sqrt{2}+2\sqrt{1}}+\frac{1}{2\sqrt{3}+3\sqrt{2}}+\frac{1}{3\sqrt{4}+4\sqrt{3}}+...+\frac{1}{2017\sqrt{2018}+2018\sqrt{2017}}\)
Bài 2: Chứng minh rằng các biểu thức sau có giá trị là số nguyên
A = \(\left(\sqrt{57}+3\sqrt{6}+\sqrt{38}+6\right)\left(\sqrt{57}-3\sqrt{6}-\sqrt{38}+6\right)\)
B = \(\frac{2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}}{\sqrt{6}+\sqrt{2}}\)
1. Chứng mình rằng:
a) \(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{5}+\sqrt{6}}+...+\frac{1}{\sqrt{47}+\sqrt{48}}>3\)
b) \(\left(\sqrt{22}-3\sqrt{2}\right).\sqrt{10+3\sqrt{11}}=2\)
2. Rút gọn các biểu thức sau:
\(A=\frac{\sqrt{\sqrt{5}+2}}{\sqrt{2}}-\sqrt{\sqrt{5}-1}\)
\(B=\left(5+\sqrt{21}\right).\left(\sqrt{14}-\sqrt{6}\right).\sqrt{5+\sqrt{21}}\)
1) Chứng minh đẳng thức \(\left(\frac{3\sqrt{2}-\sqrt{6}}{\sqrt{27}-3}-\frac{\sqrt{150}}{3}\right).\frac{1}{\sqrt{6}}=-\frac{4}{3}\)
2) Chứng minh \(\sqrt{\sqrt{3}-\sqrt{3-\sqrt{13-4\sqrt{3}}}}=1\)
\(a,\frac{1}{\sqrt{3}-\sqrt{5}}+\frac{1}{\sqrt{3}+\sqrt{5}}\)
\(b,\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{48}+\sqrt{49}}\)
a)Cho a>b>0 chứng minh rằng \(\frac{1}{a+b}\le\frac{1}{2\sqrt{ab}}\)
b) Chứng minh \(\frac{\sqrt{2}-\sqrt{1}}{3}+\frac{\sqrt{3}-\sqrt{2}}{5}+\frac{\sqrt{4}-\sqrt{3}}{7}+...+\frac{\sqrt{2011}-\sqrt{2010}}{4021}< \frac{1}{2}\)
Chứng minh A< 1. Biết
\(A=\frac{1}{1\sqrt{2}+2\sqrt{1}}+\frac{1}{2\sqrt{3}+3\sqrt{2}}+\frac{1}{3\sqrt{4}+4\sqrt{3}}+...+\frac{1}{399\sqrt{400}+400\sqrt{399}}\)
Chứng minh
\(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}.\sqrt{3}}+\frac{1}{\sqrt{3}.\sqrt{4}}+...+\frac{1}{\sqrt{2004}.\sqrt{2005}}< 2\)